Special Functions and Their Link with Nonlinear Rod Theory

2020 ◽  
pp. 21-62
Author(s):  
Giovanni Mingari Scarpello ◽  
Daniele Ritelli
Keyword(s):  
Special Functions ◽  
Rod Theory ◽  
Nonlinear Rod Theory

On the Kinematics of the Octopus’s Arm

2009 ◽  
Vol 2 (1) ◽  
Author(s):  
Yaron Levinson ◽  
Reuven Segev
Keyword(s):  
Cross Section ◽  
Muscle Fibers ◽  
Kinematic Model ◽  
Three Dimensional ◽  
Point Of View ◽  
Rod Theory ◽  
Inverse Kinematics Problem ◽  
Nonlinear Rod Theory ◽  
Configuration Control

The kinematics of the octopus’s arm is studied from the point of view of robotics. A continuum three-dimensional kinematic model of the arm, based on a nonlinear rod theory, is proposed. The model enables the calculation of the strains in various muscle fibers that are required in order to produce a given configuration of the arm—a solution to the inverse kinematics problem. The analysis of the forward kinematics problem shows that the strains in the muscle fibers at two distinct points belonging to a cross section of the arm determine the curvature and the twist of the arm at that cross section. The octopus’s arm lacks a rigid skeleton and the role of material incompressibility in enabling the configuration control is studied.

Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Static and Linearized Dynamic Examples

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Karin Nachbagauer ◽  
Peter Gruber ◽  
Johannes Gerstmayr
Keyword(s):  
Finite Element ◽  
Continuum Mechanics ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Geometrically Nonlinear ◽  
Finite Element Software ◽  
Rod Theory ◽  
Elastic Forces ◽  
Nonlinear Rod Theory ◽  
Shear Deformable

In the present paper, a three-dimensional shear deformable beam finite element is presented, which is based on the absolute nodal coordinate formulation (ANCF). The orientation of the beam’s cross section is parameterized by means of slope vectors. Both a structural mechanics based formulation of the elastic forces based on Reissner’s nonlinear rod theory, as well as a continuum mechanics based formulation for a St. Venant Kirchhoff material are presented in this paper. The performance of the proposed finite beam element is investigated by the analysis of several static and linearized dynamic problems. A comparison to results provided in the literature, to analytical solutions, and to the solution found by commercial finite element software shows high accuracy and high order of convergence, and therefore the present element has high potential for geometrically nonlinear problems.

Planar Vibration of an Elastica Arch: Theory and Experiment

1990 ◽  
Vol 112 (3) ◽  
pp. 374-379 ◽  
Author(s):  
N. C. Perkins
Keyword(s):  
Natural Frequencies ◽  
Numerical Solutions ◽  
Large Range ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Geometrically Nonlinear ◽  
Linear Vibration ◽  
Elastic Rod ◽  
Rod Theory ◽  
Nonlinear Rod Theory

This investigation examines the planar, linear vibration of a deep arch that is described by a simply supported elastica. The arch is formed from an elastic rod that buckles nonlinearly under the action of a large, steady end-load. A theoretical model is proposed that governs the planar response of the rod about a generally curved, pre-stressed equilibrium. The model utilizes a geometrically nonlinear rod theory to describe the planar bending and extension of the rod centerline. The equations of motion are linearized about an elastica equilibrium and numerical solutions for free vibration are determined using a variational formulation of the associated eigenvalue problem. Natural frequencies and mode shapes are computed over a large range of centrally and eccentrically applied end-loads. Results from an experimental modal test provide support for the model.

scholarly journals New fractional approaches for n-polynomial P-convexity with applications in special function theory

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shu-Bo Chen ◽  
Saima Rashid ◽  
Muhammad Aslam Noor ◽  
Zakia Hammouch ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Calculus ◽  
Convex Functions ◽  
Special Functions ◽  
Real Life ◽  
Fractional Integrals ◽  
Digamma Function ◽  
Novel Approach ◽  
Inequality Theory ◽  
New Strategy ◽  
Significant Mechanism

Abstract Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

scholarly journals Gibbs partitions, Riemann–Liouville fractional operators, Mittag–Leffler functions, and fragmentations derived from stable subordinators

2021 ◽  
Vol 58 (2) ◽  
pp. 314-334
Author(s):  
Man-Wai Ho ◽  
Lancelot F. James ◽  
John W. Lau
Keyword(s):  
Special Functions ◽  
Dirichlet Distribution ◽  
Random Trees ◽  
Fractional Integrals ◽  
Biased Sampling ◽  
Scaling Limits ◽  
Fractional Operators ◽  
Random Partitions ◽  
Two Parameter ◽  
Potential Use

AbstractPitman (2003), and subsequently Gnedin and Pitman (2006), showed that a large class of random partitions of the integers derived from a stable subordinator of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson–Dirichlet distribution, $\textrm{PD}(\alpha,\theta)$, whose corresponding $\alpha$-diversity/local time have generalized Mittag–Leffler distributions, denoted by $\textrm{ML}(\alpha,\theta)$. Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it relates to a study of Riemann–Liouville fractional integrals and size-biased sampling, and in decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide characterizations of general laws associated with nested families of $\textrm{PD}(\alpha,\theta)$ mass partitions that are constructed from fragmentation operations described in Dong et al. (2014). These operations are known to be related in distribution to various constructions of discrete random trees/graphs in [n], and their scaling limits. A centerpiece of our work is results related to Mittag–Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the $\textrm{ML}(\alpha,\theta)$ variables. Notably, this leads to an interpretation within the context of $\textrm{PD}(\alpha,\theta)$ laws conditioned on Poisson point process counts over intervals of scaled lengths of the $\alpha$-diversity.

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